€¦ · solution manual for developmental mathematics 1st edition by blitzer link full download

Solution Manual for Developmental Mathematics 1st Edition by Blitzer

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developmental-mathematics-1st-edition-by-blitzer/

Chapter 2

Integers and Introduction to Solving Equations

2.1 Check Points

a. −500 b.

−282

2.

a. 6 > −7 because 6 is to the right of –7 on the number line.

− 8 < −1 because –8 is to the left of –1 on the number line.

− 25 < −2 because –25 is to the left of –2 on the number line.

− 14 < 0 because –14 is to the left of 0 on the number line.

a. −8 = 8 because –8 is 8 units from 0.

6 = 6 because 6 is 6 units from 0.

− 8 = −8 because 8 is 8 units from 0 and the negative of 8 is −8.

a. The opposite of −14 is 14.

The opposite of 17 is −17.

The opposite of 0 is 0.

a. We can use the double-negative rule, − − a = a, to simplify.

− 25 = 25

We cannot use the double-negative rule when one of the negatives is inside the absolute value bars.

− 14 = −14

We can use the double-negative rule, − − a = a, inside the absolute value bars.

− 30 = 30 = 30

7. a. Take aspirin daily: 5; Blood relative 95 or older: 10; Less than 6 to 8 hours sleep: −1; Less than 12 years education: −6

Number line:

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Less than 12 years education; Less than 6 to 8 hours sleep; Take aspirin daily; Blood relative 95 or older

Copyright © 2017 Pearson Education, Inc. 61

Chapter 2 Integers and Introduction to Solving Equations

2.1 Concept and Vocabulary Check

..., 3, 2, 1,0,1,2,3,...

left

the distance from 0 to a

opposites

a

2.1 Exercise Set

–20

65

8

–12,500

–3000

–3

–4,000,000,000

-14

–2 < 7 because –2 is to the left of 7 on the number

line.

–1 < 13 because –1 is to the left of 13 on the number line.

–13 < –2 because –13 is to the left of –2 on the

number line.

–1 > –13 because –1 is to the right of –13 on the number line.

8 > –50 because 8 is to the right of –50 on the

number line.

7 > –9 because 7 is to the right of –9 on the number line.

–100 < 0 because –100 is to the left of 0 on

the number line. 0 > –300 because 0 is to the right of –300 on

the number line.

− 14 = 14

− 16 = 16

14 = 14

16 = 16

− 300,000 = 300,000

−1,000,000 = 1,000,000

− 14 = −14

− 16 = −16

The opposite of −7 is 7.




− − 70 = 70

− − 80 = 80

− − 70 = −70

− − 80 = −80

− − 12 = 12 = 12

− − 14 = 14 = 14

− − − 12 = − 12 = −12

− − − 14 = − 14 = −14

− 6 > −3 , because 6 > 3 62 Copyright © 2017 Pearson Education, Inc.

−20 < −50 , because 20 < 50

− − 6 < − −3 , because –6 < –3

− − 20 > − −50 , because –20 > –50

− − 5 > − −5 , because 5 > –5

− − 7 > − −7 , because 7 > –7

–63 has the greater absolute value because it is further from zero on the number line.

–74 has the greater absolute value because it

is further from zero on the number line.

− x = − − 5 = 5

− − x = − − 6 = 6

1873; Ulysses S. Grant 1985; Ronald Reagan Cleveland Reagan Van Buren and Cleveland Van Buren and Cleveland Grant and Reagan Grant, Kennedy, and Reagan

a.

� Rhode Island, Georgia, Louisiana, Florida, Hawaii

a.

� Wyoming, Wisconsin, Washington, West Virginia, Virginia

a. 4 F

� −4 F

Section 2.1 Introduction to Integers

ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ȁ ЀĀ Ȁ ⸀Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Ā ᜀ Yes, 4 and –4 are the same distance from 0 on opposite sides of 0 on a number line.

a. 9 F

−9 F

Yes, 9 and –9 are the same distance from 0 on opposite sides of 0 on a number line.

When the wind speed is 5 miles per hour and the air

temperature is −5 F the temperature feels like

−16 F.

When the wind speed is 50 miles per hour and the air temperature is 10 F the temperature feels like

−17 F.

It feels colder when the wind speed is 50 miles per

hour and the air temperature is 10 F . When the wind speed is 60 miles per hour and the

air temperature is 15 F the temperature feels like

−11 F.

When the wind speed is 15 miles per hour and the air temperature is 5 F the temperature feels like

−13 F.

It feels colder when the wind speed is 15 miles per

hour and the air temperature is 5 F . – 70. Answers will vary. does not make sense; Explanations will vary.

Sample explanation: The Titanic’s resting place

is higher because it is less feet below sea level. does not make sense; Explanations will vary.

Sample explanation: The lowest a class size

could be is zero. makes sense

does not make sense; Explanations will vary.

Sample explanation: We cannot use the double-

negative rule when one of the negatives is inside

the absolute value bars. true true true



false; Changes to make the statement true will vary.

A sample change is: If a > b and a and b are

integers, then a can be a negative or positive

integer depending upon the value of b.

− − 37 + − 93 = 37 + 93 = 130

− − 600 − − 76 = − − 600 − 76 = 600 − 76 = 524

3 6(9 − 5) + 2 = 3 6(4) + 2

3 24 + 2

3 26

78

6 x 2

+ 5 x + 3 = 6 2 2

+ 5 2 + 3

6 4 + 5 2 + 3

24 + 10 + 3

37

associative property of addition

814 − 347 = 467

150 + − 90 = 60

− 40 + − 10 = −50

2.2 Check Points

a. A loss of $60 followed by a loss of $40 results in a

loss of $100. − 60 + ( − 40) = −100

A gain of $60 followed by a loss of $40 results in a gain of $20. 60 + ( − 40) = 20

4 + ( − 7) = −3

Start at 4 and move 7 units to the left.

a. − 1 + ( − 3) = −4 Start at −1 and move 3 units to the left.

− 5 + 3 = −2 Start at −5 and move 3 units to the right.

a. − 7 + ( −2) = −9; Add the absolute values and

use the common sign.

− 18 + ( −46) = −64; Add the absolute values and

use the common sign.

52 + 43 = 95; Add the absolute values and use

the common sign. a. − 12 + 7 = −5; Subtract the absolute values and

use the sign of the number with the greater absolute value.

20 + −3 = 17; Subtract the absolute values and


a. − 46 + 71 = 25; Subtract the absolute values and


27 + − 95 = −68; Subtract the absolute values

and use the sign of the number with the greater absolute value.

a. 26 + −48 = −22; The addends have different

signs, so subtract the absolute values and use the

sign of the number with the greater absolute value.

− 35 + − 102 = −137; The addends have the

same signs, so add the absolute values and use the common sign.

− 453 + 619 = 166; The addends have different

signs, so subtract the absolute values and use the

sign of the number with the greater absolute value.

� � �=� � � � � � 7

9 + − 79 = 0; The sum of any integer and

its opposite is 0. − 23 + 44 + − 66 + 38 = 44 + 38 + − 23 + −66

82 + −89

−7

64 Copyright © 2017 Pearson Education, Inc.

2 + (−4) + 1+ (−5) + 3 = (2 + 1+ 3) + (−4) + (−5)

6 + (−9)

−3 At the end of 5 months the water level was down 3 feet.


a; right; left; a; sum

0

negative

negative integer

positive integer

0

positive integer

negative integer

2.2 Exercise Set

A loss of $8 followed by a loss of $2 results in a loss of $10. − 8 + (− 2) = −10

A loss of $10 followed by a loss of $6 results in a loss of $16. − 10 + (− 6) = −16

A gain of $12 followed by a loss of $8 results in a gain of $4. 12 + (−8) = 4

A gain of $15 followed by a loss of $10 results in a gain of $5. 15 + (− 10) = 5

A gain of $20 followed by a loss of $25 results in a loss of $5. 20 + (−25) = −5

A gain of $30 followed by a loss of $36 results in a loss of $6. 30 + (− 36) = −6

Section 2.2 Adding Integers

A loss of $4 followed by a gain of $5 results in a gain of $1. − 4 + 5 = 1

A loss of $7 followed by a gain of $7 results in a

gain of $1. − 6 + 7 = 1

7 + − 3 = 4

7 + − 2 = 5

− 2 + −5 = −7

− 1+ − 5 = −6

− 6 + 2 = −4

− 8 + 3 = −5

3 + −3 = 0

5 + − 5 = 0

− 8 + − 10 = −18

− 4 + −6 = −10

− 17 + −36 = −53



− 19 + − 47 = −66

17 + 36 = 53

19 + 47 = 66

− 12 + 7 = −5

− 14 + 6 = −8

15 + − 6 = 9

18 + − 11 = 7

− 46 + 93 = 47

− 37 + 82 = 45

34 + − 76 = −42

38 + − 89 = −51

− 68 + − 91 = −159

− 58 + − 83 = −141

− 247 + 913 = 666

− 358 + 817 = 459

247 + 913 = 1160

358 + 817 = 1175

247 + −247 = 0

358 + −358 = 0

− 247 + 247 = 0

− 358 + 358 = 0

4 + − 7 + − 5 = 4 + − 7 + −5

4 + −12 −8

10 + −3 + − 8 = 10 + −3 + −8

10 + −11 −1

7 + − 10 + 2 + −3 = 7 + 2 + −10 + −3

9 + −13

−4

5 + − 7 + 3 + − 6 = 5 + 3 + − 7 + −6

8 + −13 −5

− 19 + 13 + − 33 + 17 = 13 + 17 + − 19 + −33

30 + −52

−22

− 18 + 15 + − 34 + 25 = 15 + 25 + − 18 + −34

40 + −52

−12 27 + − 13 + 14 + −28 = 27 + 14 + − 13 + −28

= 41+ −41

= 0

38 + −16 + 11+ − 33 = 38 + 11 + −16 + −33

= 49 + −49

= 0

15 + − 63 = −48

11+ − 74 = −63

− 50 + 13 = −37

− 40 + 17 = −23

− 26 + 39 = −13

− 37 + 54 = −17

− 3 + −5 + 2 + − 6 = −8 + −4

8 + 4

12

4 + −11 + − 3 + − 4 = −7 + −7

7 + 7

14


− 20 + − 15 + − 25 = −20 + − −10

−20 + −10

−30

− 25 + − 18 + − 26 = − 25 + − −8

− 25 + −8

−33

Left side: 6 + 2 + −13 = 6 + − 11 = −5

Right side: − 3 + 4 + − 8 = −3 + − 4 = −7

+ 2 + −13 > −3 + 4 + −8 because − 5 > −7

Left side: − 8 + −6 − 10 = −14 − 10 = −24

= −8 + −12 = −20 Right side: − 8 + 9 + −21

− 10 < −8 +

− 8 + −6 9 + −21 because

− 24 < −20 − 56 + 100 = 44

The high temperature was 44°F. − 4 + 49 = 45

The high temperature was 45°F. − 1312 + 712 = −600

The elevation of the person is 600 feet below sea level.

− 512 + 642 = 130

The elevation of the person is 130 feet above sea level.

− 7 + 15 − 5 = 3

The temperature at 4:00 P.M. was 3°F. − 15 + 13 + ( −4) = −6

The team had a total loss of 6 yards. 27 + 4 − 2 + 8 − 12

27 + 4 + 8 + −2 − 12

34 − 14

25 The location of the football at the end of the fourth play is at the 25-yard line.

Section 2.2 Adding Integers

20 + 3 + ( − 2) + ( − 1) + ( − 4) + 2

20 + 3 + 2 + − 2 + ( −1) + ( −4)

25 − 7

18 The level of the reservoir is 18 feet.

a. 2304 + − 3603 = −1299

The deficit is $1299 billion for 2011.

2450 + − 3537 = −1087

The deficit is $1087 billion for 2012. This is better than 2011 because 2012 had less debt.

− 1299 + −1087 = −2386

The combined deficit is $2386 billion.

a. 2105 + − 3518 = −1413

The deficit is $1413 billion for 2009.

2163 + −3457 = −1294

The deficit is $1294 billion for 2010. This is better than 2009 because 2010 had less debt.

− 1413 + − 1294 = −2707

The combined deficit is $2707 billion. – 76. Answers will vary. makes sense makes sense does not make sense; Explanations will vary.

Sample explanation: The sum of two

negative integers is a negative integer. makes sense true false; Changes to make the statement true will vary.

A sample change is: The sum of a positive integer

and a negative integer can be positive or negative

or zero. true false; Changes to make the statement true will

vary. The absolute value of two negative integers is always a positive integer.

The sum is negative. The sum of two negative

numbers is always a negative number.



The sum is negative. When finding the sum of

numbers with different signs, use the sign of the

number with the greater absolute value as the sign

of the sum. Since a is further from 0 than c, we

use a negative sign.

The sum is positive. When finding the sum of

numbers with different signs, use the sign of the

number with the greater absolute value as the sign

of the sum. Since c is further from 0 than b, we

use a positive sign.

Though the sum inside the absolute value is

negative, the absolute value of this sum is positive.

– 90. Answers will vary.

2 ⋅ 3 2 − 2 ⋅ 3

2 = 6

2 − 2 ⋅ 9 = 36 − 18 = 18

2 y + 7 = 13

2 3 + 7 = 13

6 + 7 = 13

13 = 13, true

The number is a solution.

commutative property of addition

7 − 10 = 7 + (− 10) = −3

− 8 − 13 = − 8 + (− 13) = −21

− 8 − (− 13) = −8 + 13 = 5

2.3 Check Points Note that for Check Points #1 – 2, first change all

subtractions to additions of opposites.

a. 3 − 11 = 3 + (− 11) = −8

4 − (−5) = 4 + 5 = 9

− 7 − (−2) = −7 + 2 = −5

a. − 46 − 87 = − 46 + (− 87) = −133

129 − − 317 = 129 + 317 = 446

− 164 − −38 = − 164 + 38 = −126

First change all subtractions to additions of opposites. Then add the positive numbers

and negative numbers separately. 10 − (− 12) − 4 − (−3) − 6 = 10 + 12 + (− 4) + 3 + (−6)

(10 + 12 + 3) + (− 4) + (−6)

25 + (−10)

15

The difference can be found by subtracting the

elevation of the Marianas Trench from the elevation of Mount Everest.

8848 − −10,915 = 8848 + 10,915 = 19,763

The difference in elevation is 19,763 meters.

a. 3 − −5 = 3 + 5 = 8

The difference in lifespan is 8 years.

− 1− −15 = −1+ 15 = 14

The difference in lifespan is 14 years.

2 + − 5 = −3

You shrink your lifespan by 3 years.

5 + − 5 = 0

There is no change to your lifespan.


−14

14

14

− 8; −14

3; − 12 ; −23

2.3 Exercise Set

a. −12

5 − 12 = 5 + −12

a. −10

4 − 10 = 4 + −10


a. 7

5 − −7 = 5 + 7

a. 8

2 − −8 = 2 + 8

14 − 8 = 14 + −8 = 6

15 − 2 = 15 + −2 = 13

8 − 14 = 8 + −14 = −6

2 − 15 = 2 + (− 15) = −13

3 − − 20 = 3 + 20 = 23

5 − −17 = 5 + 17 = 22

− 7 − −18 = − 7 + 18 = 11

− 5 − −19 = −5 + 19 = 14

− 13 − − 2 = −13 + 2 = −11

− 21− − 3 = − 21+ 3 = −18

− 21− 17 = −21+ − 17 = −38

− 29 − 21 = −29 + − 21 = −50

− 45 − −45 = −45 + 45 = 0

− 65 − −65 = −65 + 65 = 0

23 − 23 = 23 + − 23 = 0

26 − 26 = 26 + −26 = 0

13 − −13 = 13 + 13 = 26

15 − − 15 = 15 + 15 = 30

0 − 13 = 0 + − 13 = −13

0 − 15 = 0 + − 15 = −15

Section 2.3 Subtracting Integers

25. 0 − −13 = 0 + 13 = 13

26. 0 − − 15 = 0 + 15 = 15

27. − 29 − 86 = − 29 + (− 86) = −115

28. − 37 − 95 = − 37 + (−95) = −132

29. 274 −

− 391 = 274 + 391 = 665

30. 268 − −419 = 268 + 419 = 687

31. − 146 − −89 = −146 + 89 = −57

32. − 263 − − 98 = −263 + 98 = −165

33. − 146 − −146 = − 146 + 146 = 0

34. − 263 − − 263 = − 263 + 263 = 0

35. − 146 − 146 = − 146 + − 146 = −292

36. − 263 − 263 = − 263 + − 263 = −526

37. 13 − 2 − − 8 = 13 + − 2 + 8

+ 8

+

−2 = 13

= 21+ −2 = 19

38. 14 − 3 − −7 = 14 + −3 + 7

+ 7

+

−3 = 14

= 21+ −3 = 18

39. 9 − 8 + 3 − 7 = 9 + −8 + 3 + −7 = 9 + 3 + −8 + −7

= 12 + −15 = −3

40. 8 − 2 + 5 − 13 = 8 + −2 + 5 + −13 = 8 + 5 + − 2 + −13

= 13 + −15 = −2



− 6 − 2 + 3 − 10

− 6 + − 2 + 3 + −10 −6 + −2 + −10 + 3

− 18 + 3

−15

− 9 − 5 + 4 − 17

− 9 + − 5 + 4 + −17 −9 + −5 + − 17 + 4

− 31+ 4

−27

− 10 − − 5 + 7 − 2

− 10 + 5 + 7 + −2 −10 + − 2 + 5 + 7

− 12 + 12

0

− 6 − − 3 + 8 − 11

− 6 + 3 + 8 + −11 −6 + −11 + 3 + 8

− 17 + 11

−6

− 23 − 11− − 7 + −25

−23 + −11 + 7 + −25 −23 + −11 + −25 + 7

− 59 + 7

−52

− 19 − 8 − − 6 − −21

( −19) + (− 8) + 6 − (−21)

(− 19) + (−8) + 6 + (+21) − 27 + 27

0

20 − 37 + 19 − 48 − −17

20 + − 37 + 19 + −48 + 17 −37 + −48 + 20 + 19 + 17

− 85 + 56

−29

17 − 42 + 11− 78 − −13

17 + −42 + 11+ − 78 + 13 −42 + − 78 + 17 + 11+ 13

− 120 + 41

−79

− 823 − 146 − 50 − −832

− 823 + − 146 + − 50 + 832 −823 + −146 + − 50 + 832

− 1019 + 832

−187

− 726 − 422 − 921− −816

− 726 + −422 + − 921 + 816 −726 + − 422 + −921 + 816

− 2069 + 816

−1253

15 − − 17 = 15 + 17 = 32

29 − − 11 = 29 + 11 = 40

− 5000 − − 7 = − 5000 + 7 = −4993

− 6000 − − 8 = −6000 + 8 = −5992

18 − − 4 = 18 + 4 = 22

16 − − 5 = 16 + 5 = 21

− 100 − 40 = −100 + −40 = −140

− 100 − 60 = −100 + − 60 = −160

− 50 − 20 = −50 + −20 = −70

− 70 − 20 = −70 + −20 = −90

− 534 − 100 = − 534 + − 100 = −634

− 342 − 100 = − 342 + − 100 = −442

− 760 − 40 = −760 + −40 = −800

− 540 − 60 = −540 + − 60 = −600


Left side: − 26 − − 18 = −26 + 18 = −8

Right side: − 60 − −48 = −60 + 48 = −12

− 26 − − 18 > −60 − −48 because − 8 > −12

Left side: − 26 − 51 = − 26 + −51 = −77

Right side: − 44 − 27 = − 44 + −27 = −71

− 26 − 51 < − 44 − 27 because − 77 < −71

12 − x − y = 12 − − 2 − 5

12 + 2 + −5

14 + −5

9

15 − x − y = 15 − −3 − 7

15 + 3 + −7

18 + −7

11 9 − x = 13

− − 4 = 13

9 + 4 = 13

13 = 13, true

The number is a solution. 8 − x = 15

8 − −7 = 15

8 + 7 = 15

15 = 15, true


− 3 − 6 − 10 = − 3 − 6 + −10

− 3 − −4

− 3 + 4

1

− 5 − 4 − 12 = − 5 − 4 + −12

− 5 − −8

− 5 + 8

3 Elevation of Mount McKinley – elevation of

Death Valley

20,320 − −282 = 20,602 The difference in elevation between the two geographic locations is 20,602 feet.

Section 2.3 Subtracting Integers

Elevation of Mount Kilimanjaro – elevation of Qattara Depression

19,321− − 436 = 19,757 The difference in elevation between the two geographic locations is 19,757 feet.

2 − −19 = 2 + 19 =

21 The difference is 21 F.

6 − − 12 = 6 + 12 =

18 The difference is 18 F.

− 19 − − 22 = − 19 +

22 = 3 3 F warmer

− 12 − − 19 = − 12 +

19 = 7 7 F warmer

− 1413 + − 1300 = −2713


− 161+ −1413 + − 1300 = −2874


128 − −1300 = 128 + 1300 =

1428 The difference is $1428 billion.

128 − −1413 = 128 + 1413 =

1541 The difference is $1541 billion. – 85. Answers will vary. makes sense makes sense makes sense makes sense true false; Changes to make the statement true will

vary. A sample change is: 7 − − 2 = 9.

true true positive negative



negative

positive

98. x − y = − 6 − − 8 = − 6 + 8 = 2 =

2 x − y = −6 − − 8 = 6 − 8 = 6 + −8 =

−2 x + y = −6 + −8 = − 14 = 14

From least to greatest is: x − y ; x − y ; x + y

– 100. Answers will vary. seventy-six thousand, three hundred five

203

47 9541

94 14 0

141 141

0 9541÷ 47 = 203

70 + 84 + 90 + 91+ 100 = 435 = 87 5

4(− 3) = (− 3) + (− 3) + (−3) + (−3) = −12

3(−3) = (− 3) + (− 3) + (−3) = −9

2( − 3) = −6

1( − 3) = −3

0( − 3) = 0

−1( −3) = 3

− 2( −3) = 6

− 3( − 3) = 9

− 4( −3) = 12

2.4 Check Points

a. 8( −5) = −40 The product of two integers with

different signs is negative.

− 6 2 = −12 The product of two

integers with different signs is negative.

a. − 7 −10 = 70 The product of two

integers with same signs is positive.

− 4 − 9 = 36 The product of two

integers with same signs is positive. a. 15( − 19) = −285 The product of two integers

with different signs is negative.

b. − 6 − 204 = 1224 The product of

two integers with same signs is positive.

− 204 0 = 0 The product of any number

and zero is zero.

− 7 38 = −266 The product of two

integers with different signs is negative. a. ( − 2)(3)( − 1)(4) = 24 When multiplying an even

number of negative integers the product is positive.

( − 1)( − 3)(2)( −1)(5) = −30 … When multiplying

an odd number of negative integers the product is negative.

a. − 6 2

= −6 −6 = 36

− 6 2 = − 6 ⋅ 6 = −36

− 5 3 = −5 −5 −5 = −125

− 1 4 = − 1 −1 − 1 − 1 = 1

− 14 = − 1 ⋅ 1 ⋅ 1 ⋅ 1 = −1

− 2 5 = − 2 −2 −2 −2 − 2 = −32

a. −45

= −9 The quotient of two integers with

5 different signs is negative.

− 30 ÷ − 10 = 3 The quotient of two

integers with same signs is positive.

1220

= −305 The quotient of two integers with

−4 different signs is negative.

d.

0

= 0Any nonzero integer divided into −1220 zero is zero.


a. Because there are 7 days in a week, we can find

the number of deaths in a week by multiplying

the number of daily deaths by 7.

− 156, 000 ⋅ 7 = −1, 092, 000 Each week, there are 1,092,000 deaths in the world.

Because there are 7 days in a week, we can

find the number of births in a week by

multiplying the number of daily births by 7. 384, 000 ⋅ 7 = 2, 688, 000 Each week, there are 2,688,000 births in the world.

We find the increase in world population

each week by combining the number of deaths and the number of births.

− 1, 092, 000 + 2, 688, 000 = 1, 596, 000 Each week, the population increases by 1,596,000.


negative

positive

positive

negative

0

negative

positive

0

undefined

2.4 Exercise Set

5( − 9) = −45

10( −7) = −70

− 8 −3 = 24

− 9 − 5 = 45

− 3 7 = −21

Section 2.4 Multiplying and Dividing Integers

− 4 8 = −32

− 19 − 1 = 19

− 11 −1 = 11

0( −19) = 0

0( −11) = 0

12( −13) = −156

13( − 14) = −182

− 6 − 207 = 1242

− 6 − 308 = 1848

− 207 0 = 0

− 308 0 = 0

( − 5)( − 2)(3) = 30

( − 6)( −3)(10) = 180

( − 4)( −3)( −1)(6) = −72

( − 2)( −7)( −1)(3) = −42

− 2( − 3)( − 4)( −1) = 24

− 3( −2)( − 5)( −1) = 30

( − 3)( − 3)( −3) = −27

( − 4)( − 4)( −4) = −64

5( − 3)( − 1)(2) 3 = 90

2( − 5)( − 2)(3) 1 = 60

( − 2)( −2)( − 2)( −2)( − 2) = −32

( − 2)( −2)( − 2)( − 2)( − 2) − 2 = 64

( − 8)( − 4)(0)( − 17)( − 6) = 0

( − 9)( −12)( −18)(0)( −3) = 0



31. − 4 2 = − 4 − 4 = 16

32. − 7 2 = −7 − 7 = 49

33. − 4 2 = − 4 ⋅ 4 = −16

34. − 7 2 = − 7 ⋅ 7 = −49

35. − 10 3 = − 10 −10 −10 = −1000

36. − 4 3 = − 4 − 4 −4 = −64

37. − 103 = − 10 ⋅ 10 ⋅ 10

= −1000

38. − 43 = − 4 ⋅ 4 ⋅ 4 = −64

39. − 3 4 = − 3 −3 − 3 −3 = 81

40. − 2 4 = − 2 −2 −2 −2 = 16

41. − 34 = − 3 ⋅ 3 ⋅ 3 ⋅ 3 = −81

42. − 2 4 = − 2 ⋅ 2 ⋅ 2 ⋅ 2 = −16

5 =

43. − 1 −1 − 1 − 1 −1 − 1 = −1

6 =

44. − 1 −1 − 1 − 1 −1 − 1 − 1 = 1

45. − 18 = − 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 = −1

46. − 1

10 = − 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 = −1

47. 12 = −3

−4

40

= −8

−5

−21

= −7 3

−27

= −9 3

− 60 ÷ 6 = −10

52. − 80 ÷ 8 = −10

53. − 180 ÷ −30 = 6

54. − 120 ÷ −20 = 6

55. 120 = −12 −10

56. 130 = −13 −10

57. 0 ÷ − 120 = 0

58. 0 ÷ − 130 = 0

59. − 120 ÷ 0 undefined

60. − 130 ÷ 0 undefined

61. −3542 = 506

−7

−2448

= 408

−6

− 234 ÷ 13 = −18

− 304 ÷ 16 = −19

743 ÷ −743 = −1

971 ÷ −971 = −1

− 73 −4 = 292

− 96 − 5 = 480

2 − 15 = −30

3 − 12 = −36

− 5 2 = − 5 − 5 = 25

− 6 2 = −6 −6 = 36

− 5 2 = − 5 ⋅ 5 = −25


74. − 6 2 = − 6 ⋅ 6 = −36

75. 460 = −92 −5

696

= −87

−8

−44

= 4 −11

−84

= 7 −12

Left side: − 8 5 = −40

Right side:18 ÷ −2 = −9

− 8 5 < 18 ÷ −2 because − 40 < −9

Left side: − 3 15 = −45

Right side: 60 ÷ −5 = −12

− 3 15 < 60 ÷ −5 because − 45 < −12

81.

−48

=

− 48

=

−48

= −8 xy − 2 −3 6

82.

−75

=

−75

=

−75

= −5 xy − 3 −5 15

83. −30 + 5 = 5x

x

−30 + 5 = 5 −3

−3

10 + 5 = −15

15 = −15, false

The number is not a solution.

−70

+ 32 = 6x x

−70 + 32 = 6 −7

−7 10 + 32 = −42

42 = −42, false


6 − 3 = −18

The decrease is 18 .

Section 2.4 Multiplying and Dividing Integers

7 − 4 = −28

The decrease is 28 .

65 29 − 35 = 65 29 + −35 = 65 − 6 =

−390 The total loss is $390.

85 39 − 47 = 85 39 + −47 = 85 − 8

= −680 The total loss is $680.

a. 16 −5 = −80

You are fined 80 cents.

10 8 = 80

You are owed 80 cents.

Neither owes money to the other because the values are opposites.

a. 12 −5 = −60

You are fined 60 cents.

16 8 = 128

You are owed 128 cents.

− 60 + 128 = 68

We owe you 68 cents.

−32, 200

= −4600 7

Each person losses $4600.

−26,500

= −5300 5

Each person losses $5300.

− 311 + − 330 + − 357 + − 344 + −338 5

−1680

= −336 5

On average, women with college degrees get

paid $336 less per week than their male counterparts during this five-year period.

−344 + −338 = −682 = −341 2 2

On average, women with college degrees get paid $341 less per week than their male

counterparts during this two-year period. – 100. Answers will vary.



does not make sense; Explanations will

vary. Sample explanation: The sign rules for multiplication and division are the same.

does not make sense; Explanations will vary.

Sample explanation: When I multiply more than two integers, I determine the sign of the product by

counting the number of negative factors. makes sense

makes sense

false; Changes to make the statement true will

vary. A sample change is: The addition of two

negative numbers results in a negative answer.

While, the multiplication of two negative integers

results in a positive answer. true

false; Changes to make the statement true will

vary. A sample change is: 0 ÷ − 154, 293 = 0.

true

− x + 6 x = − 15 + 25

x

− − 5 + 6 −5 = − 15 + 25 −5

5 + −30 = − 15 + −5 − 25 = −20, false The number is not a solution.

−12x −25,000

x


− 27 + −3 = −30

− 27 − − 3 = − 27 + 3 = −24

− 27 ÷ −3 = 9

8 + 27 ÷ 3 = 8 + 9

17

12 ÷ 4 ⋅ 23

= 12 ÷ 4 ⋅ 8

3⋅ 8

24

3 6 + 211− 8 2

= 3

6 + 2 32

3 6 + 2 ⋅ 9

3 6 + 18

3 24

72

Mid-Chapter Check Point – Chapter 2

− 80 ÷ 10 = −8

17 − −12 = 17 + 12 = 29

− 14 + − 16 = −30

− 43 = − 4 − 4 −4 = −64

− 6 −11 = 66

− 10 + 4 + − 15 + 7 = − 10 + − 15 + 4 + 7

= −25 + 11

= −14

7. − 20 ÷ − 4 = 5

8. 3 − 5 −2 − 1 = −30

9. − 15 − 19 = −15 + − 19 = −34

10. − 2 4

= − 2 −2 −2 −2 = 16

11. − 24 = − 2 ⋅ 2 ⋅ 2 ⋅ 2 = −16

12. 222 = −37 −6

− 7 − −11 − 5 + 18 = −7 + 11+ − 5 + 18

−7 + − 5 + 11+ 18

− 12 + 29

17

− 3 > −300

− 12 = 12


− 12 = −12

− −12 = 12

− − 12 = −12

15,400 − − 760 = 15,400 + 760 =

16,160 The difference is 16,160 feet.

5 + − 8 + 2 + 3 + − 7 + −1

5 + 2 + 3 + −8 + −7 + −1

10 + −16 −6 The temperature at the final measurement is −6 .

a. −13 − −9 = −13 + 9 = −4

It was 4 F colder in Wisconsin

−13 + −9 = −22 = −11 2 2

The mean was −11 F.

−13 + 15 = 2 The temperature was 2 F.

2.5 Check Points

There are no grouping symbols or exponents. We start by performing the division.

3 + −25 ÷ 5 = 3 + −5 −2

There are no grouping symbols or exponents. We

begin by evaluating the exponential expression.

÷10 −33 = 20 ÷10 −27

2 −27

−54

Section 2.5 Order of Operations with Integers; Mathematical Models

Because grouping symbols appear, we perform the operations in parentheses first.

− 20 − 3 − 7 ⋅ 2 = − 20 − 3 − 14

− 20 − −11

− 20 + 11

−9

There are no grouping symbols or exponents. We

begin by evaluating the exponential expression.

21− 25 ÷ 5 −3 2

− 7 = 21− 25 ÷ 5 9 −

7 = 21− 5 9 − 7

= 21− 45 − 7

= 21+ −45 + −7

= 21+ −52 = −31

Begin by performing the operation within the

innermost grouping symbol, the parentheses, first.

−3

− 10 4 − 62

= −3

− 10 −22

8 8

= −3

8 − 10 4

= −3 8 − 40

= −3

8

+ −40

= −3 −32

= 96

Fraction bars are grouping symbols that separate expressions into two parts. Simplify

above and below the fraction bar separately.

−68 ÷ 2 + 4 = −34 + 4

4943−7243 −30

−6

5



Because absolute value donates grouping, we perform the operations inside the absolute value bars first.

43 + − 18 − 4 − 2 − 72 − 2

3 = 43 + − 18 − −8 − 7

2 − 2

3

−43 + − 18 + 8 − 72 − 2

3

43 + − 10 − 72

− 23

43 + 10 − 72

− 23

43 + 10 − 49 − 8

43 + 10 + − 49 + −8

53 + −57

−4

Begin by substituting –5 for each occurrence of x in the algebraic expression. Then use the order of operations to evaluate the expression.

x2

− 9x + 4 x + 2 = − − 5 2

− 9 −5 + 4 −5 + 2

− − 5 2 − 9 −5 + 4 −3

− 25 − 9 −5 + 4 −3

− 25 − − 45 + −12

− 25 + 45 + −12

45 + −37

8

Begin by substituting the values for each variable in the algebraic expression. Then use the order of operations to evaluate the expression.

b2

− 4ac = −6 2

− 4 5 −2

36 − 4 5 −2

36 − −40

36 + 40

76

a. To determine whether –9 is a solution to the equation, substitute –9 for all occurrences of the variable and evaluate

each side of the equation. − 7x = 99 + 4x

− 7 − 9 = 99 + 4 −9

63 = 99 + −36

63 = 63, true The number is a solution.



To determine whether –3 is a solution to the equation, substitute –3 for all occurrences of the variable and evaluate each side of the equation.

− 5t2 + 8t + 70 = 0

− 5 −3 2 + 8 − 3 + 70 = 0

− 5 9 + 8 − 3 + 70 = 0

− 45 + −24 + 70 = 0

−69 + 70 = 0 1 = 0, false


11. a. Begin with the median weekly earnings of male college graduates in 2005. Because 2005 is 5 years after 2000, we

substitute 5 for x in the formula for earnings of males.

M = − 2x2

+ 170x + 5115 ÷ 5

= = − 2 ⋅ 52 + 170 ⋅ 5 + 5115 ÷ 5

= = − 2 ⋅ 25 + 170 ⋅ 5 + 5115 ÷ 5

= = − 50 + 850 + 5115 ÷ 5

= − 50 + 5965 ÷ 5

= 5915 ÷ 5

= 1183 The formula indicates that the median weekly earnings of male college graduates in 2005 were $1183. The bar graph shows earnings of $1167. Thus, the model overestimates weekly earnings by $1183 – $1167, or by $16.

b. Begin with the median weekly earnings of female college graduates in 2005. Because 2005 is 5 years after 2000, we

substitute 5 for x in the formula for earnings of females.

F = − 3x2

+ 145x + 3775 ÷ 5

= − 3⋅ 52 + 145⋅ 5 + 3775 ÷ 5

= − 3⋅ 25 + 145⋅ 5 + 3775 ÷ 5

= − 75 + 725 + 3775 ÷ 5

= = − 75 + 4500 ÷ 5

= = 4425 ÷ 5

= 885 The formula indicates that the median weekly earnings of female college graduates in 2005 were $885. The bar graph shows earnings of $883. Thus, the model overestimates weekly earnings by $885 – $883, or by $2.

c. Make a Table:

Male Earnings Female Earnings Difference (Gap)

in 2005 in 2005

Mathematical Models $1183 $885 $1183 − $885 = $298

Data from Figure $1167 $883 $1167 − $883 = $284




= divide

= multiply

= subtract

= add

= subtract

2.5 Exercise Set

− 6 + 5 −3 = −6 + −15

−21

− 8 + 4 − 3 = − 8 + −12

−20

7 + − 20 ÷ 4 = 7 + −5

. 2

9 + −35 ÷ 5 = 9 + −7

2

4 − 23 = 4 −8

−32

3 − 25 = 3 −32

−96

50 ÷ −10 − 3 2

= 50 ÷ −10 9

−5 9

−45

30 ÷ − 10 − 4 2

= 30 ÷ −1016

−316

−48

4 − 3 2 − 7 = 4 − 3 −5

4 − −15

4 + 15

19

7 − 3 4 − 10 = 7 − 3 −6

7 − −18

7 + 18

25

4 − 3 2 − 7 = 1 −5

−5

7 − 3 4 − 10 = 4 −6

−24

− 25 − 2 − 4 ⋅ 3 = − 25 − 2 − 12

− 25 − −10

− 25 + 10

−15

− 35 − 3 − 4 ⋅ 2 = −35 − 3 − 8

−35 − −5

−35 + 5

−30

6 −23 + 12 ÷ −4 = 6 −8 + 12 ÷ −4

. − 48 + −3

. −51

4 − 23 + 30 ÷ − 5 = 4 − 8 + 30 ÷ −5

− 32 + −6

−38

30 − 35 ÷ 5 − 3 2

− 6 = 30 − 35 ÷ 5 9 − 6

30 − 7 9 − 6

30 − 63 − 6

30 + − 63 + −6

30 + −69

−39

40 − 50 ÷ 10 − 2 2 − 8 = 40 − 50 ÷ 10 4 − 8

40 − 5 4 − 8

40 − 20 − 8

40 + −20 + −8

40 + −28

−12


− 20 − 40 ÷ 10 −4 2 + 3 = − 20 − 40 ÷ 1016 + 3

− 20 − 416 + 3

− 20 − 64 + 3

− 20 + − 64 + 3

− 84 + 3

−81

− 12 − 30 ÷ 10 − 5 2 + 6 = −12 − 30 ÷ 10 25 + 6

−12 − 3 25 + 6

−12 − 75 + 6

−12 + −75 + 6

−87 + 6

−81

− 8 + 18 2 ÷ 8 − 93 = 10 2 ÷ −13

100 ÷ −1

−100

− 10 + 4 2 ÷ 6 − 73 = − 6 2 ÷ −13

36 ÷ −1

−36

32

− 123 = 9 − 123

−33

−27

12 − 42

3 = 12 − 163

−43

−64

− 18 ÷ 3 − − 14 ÷ 7 = −6 − −2

−6 + 2

−4

− 24 ÷ 3 − − 15 ÷ 5 = − 8 − −3

− 8 + 3

−5

− 50 ÷ −5 ÷ −2 ⋅ 3 = 10 ÷ −2 ⋅ 3

− 5 ⋅ 3

−15


− 40 ÷ −5 ÷ − 2 ⋅ 4 = 8 ÷ −2 ⋅ 4

− 4 ⋅ 4

−16

2 − 3 − 9 ÷ 3 = 2 − 3 −3

2 9

18

2 − 5 − 12 ÷ 3 = 2 −5 −4

= 2 20

= 40

31. − 2

− 2 10 − 15

2 = −2

− 2 −5

2 40 40

= −2

−

40 2 25

= −2 40 − 50

= −2 −10

= 20

32. − 3

− 2 15 − 20 2

= −3

− 2 −52

30 30

= −3

− 2 25 30

= −3 30 − 50

= −3 −20

= 60

33. −50 ÷ 2 + 4 = −25 + 4

−7

9 − 16

=

−21

−7

3

−90 ÷ 3 + 6 = − 30 + 6

8 − 14−6 −24 −6

= 4

35. −3 2 + 4 − 6 = 9 + 4 −6

8 − 5 23 − 5

= 9 + −24

3

= −15 3

= −5



− 6 2 + 8 − 5 = 36 + 8 −5 3 − 68 − 62

=

36 + −40

2

=

−4

2

= −2

37. −24 − 3 − 5 = − 24 − −15 1− −2

1+ 2

= − 24 + 15

3

= −9 3

= −3

38. −32 − 2 − 6 = −32 − −12

2 − − 2

2 + 2

= −32 + 12

4

= −20 4

−5

− 36 ÷ −6 + −6 = −36 ÷ −12

−36 ÷12

−3

− 40 ÷ − 4 + −4 = −40 ÷ −8

−40 ÷ 8

−5

− 4 3 − 8 − 4 −5 = −4 − 5 − 4 −5

−4 5 − 4 −5

−20 − −20

−20 + 20

0

− 5 4 − 10 − 5 −6 = − 5 − 6 − 5 −6

− 5 6 − 5 −6

− 30 − −30

− 30 + 30

0

23

− 12 − 42

− −11 = 8 − 12 − 42

− −11

−4 − 42 − −11

4 − 16 − −11

4 + −16 + 11

15 + −16

−1

23

− 10 − 52

− −6 = 8 − 10 − 52

− −6

−2 − 52 − −6

2 − 25 − −6

2 + −25 + 6

8 + −25

−17



45 + − 12 − 2 − 3 − 82

+ −5 2

= 45 + −12 − − 6 − 82

+ −52

45 + −12 + 6 − 82

+ −52

45 + −6 − 82

+ −52

45 + 6 − 82

+ −52 45 + 6 − 64 + 25

76 + −64

12

40 + −10 − 4 − 3 − 62

+ −4 2

= 40 + −10 − − 12 − 62

+ −42

40 + −10 + 12 − 62

+ −42

40 + 2 − 62

+ −42

40 + 2 − 62

+ −42 40 + 2 − 36 + 16

58 + −36

22

x2

− 3x − 7 = −5 2

− 3 − 5 − 7

25 − 3 −5 − 7

25 − −15 − 7

25 + 15 + −7

40 + −7

33

x2

− 7x − 8 = −6 2

− 7 −6 − 8

36 − 7 −6 − 8

36 − − 42 − 8

36 + 42 + −8

78 + −8

70

− 3y2 + 7 y + 15 = − 3 − 4 2 + 7 − 4 + 15

− 316 + 7 − 4 + 15

− 48 + − 28 + 15

− 76 + 15

−61



− 4 y2 + 6 y + 23 = −4 − 3 2 + 6 −3 + 23

−4 9 + 6 − 3 + 23

−36 + −18 + 23

−54 + 23

−31

− m2 − 3m + 4 m + 5 = − − 7 2 − 3 − 7 + 4 − 7 + 5

− − 7 2 − 3 − 7 + 4 −2

− 49 − 3 −7 + 4 −2

− 49 − −21 + −8

− 49 + 21+ −8

− 57 + 21

−36

− m2 − 5m + 6 m + 2 = − − 8 2 − 5 −8 + 6 − 8 + 2

− − 8 2 − 5 −8 + 6 −6

− 64 − 5 −8 + 6 −6

− 64 − −40 + −36

− 64 + 40 + −36

− 100 + 40

−60

− 12,500x2 + 3472x + 1776 = −12,500 0 2 + 3472 0 + 1776

0 + 0 + 1776

1776

− 14,700x2 + 4723x + 1812 = − 14,700 0 2 + 4723 0 + 1812

0 + 0 + 1812

1812

55.

x + 3

− 7 =

−10 + 3

− 7

x3

+ 5x2 − 103 + 5 −102

=

− 7

− 7

− 1000 + 5 100

=

7 − 7

− 1000 + 500

=

0

−500

= 0



56.

x + 4

− 6 =

−10 + 4

− 6

x3

+ 3x2 − 103 + 3 −102

=

− 6

− 6

− 1000 + 3 100

=

6 − 6

− 1000 + 300

0

−700

0

5x − 3y + 17 = 5 − 6 − 3 4 + 17

− 30 − 12 + 17

− 42 + 17

−25

4x − 7 y + 19 = 4 − 9 − 7 2 + 19

− 36 − 14 + 19

− 50 + 19

−31

− 3m3 − m + n

2 + 5n = −3 − 23 − − 2 + − 10 2 + 5 −10

( −3 − 8 − −2 + 100 + 5 −10

( 24 − −2 + 100 + −50

( 24 + 2 + 100 + −50

( 126 + −50

( 76

− 2m3 − m + n

2 + 6n = −2 − 33 − − 3 + −8 2 + 6 −8

−2 − 27 − − 3 + 64 + 6 −8

54 − − 3 + 64 + −48

54 + 3 + 64 + −48

121+ −48

73

− 2c + a2 − 4b = −2 6 + 8 2 − 4 20

−2 6 + 64 − 80

−2 6 + −16

−12 + 16

4



a2

92

68.

− 2x = 5x + 28

62. − 5c + − 3b = −5 10 + − 3 30

= −5 10

+

81− 90

− 2 −4 = 5 − 4 + 28

8 = − 20 + 28

= −5 10

+

−9

8 = 8, true

= −50 + 9 The number is a solution.

= −41 69.

− 4x + 10 = −2

b2

− 4ac = 6 2

− 4 3 −5 3x + 1

63. − 4 −6 + 10 = −2 3 −6 + 1 = 36 − 4 3 −5

24 + 10 = −2 − 18 +

= 36 − −60 1

34 = −2 −17 = 36 + 60 34 = 34, true

= 96


64. b2

− 4ac = 3 2

− 4 6 −4 70. −3x + 1 = − 2 4x + 2

= 9 − 4 6 −4

− 3 − 1 + 1 = − 2 4 −1 + 2

= 9 − −96 3 + 1 = − 2 −4 + 2 = 9 + 96 4 = − 2 −2 = 105

4 = 4, true

b2 −122 The number is a solution.

65. =

w

w

2c − a 2 8 − 25 71.

− 7 = 5 −

= 144

5

3

−15

− 7 = 5 −

−15

16 − 25

=

144 5 3

− 3 − 7 = 5 − −5

−9

= −16 − 3 + − 7 = 5 + 5

b2

−152 −10 = 10, false 66. The number is not a solution. =

2c − a 2 7 − 29 w

w

= 225 72. − 6 = 4 −

8

2

14 − 29

− 16

−16

= 225 − 6 = 4 −

8

2

−15

− 2 − 6 = 4 − −8

= −15

67.

2x = 48 + 6x − 2 + −6 = 4 + 8

−8 = 12,

false

2 − 12 = 48 + 6 −12


− 24 = 48 + −72

73.

−3y + 6 + 5y = 7 y − 8y

− 24 = −24, true

− 3 − 2 + 6 + 5 − 2 = 7 − 2 − 8 −2 The number is a solution.

6 + 6 + − 10 = − − 14 − −16

12 + − 10 = −14 + 16

2 = 2, true



4y − 8 − y = 10 y − 3y

4 − 2 − 8 − − 2 = 10 −2 − 3 −2

− 8 − 8 − − 2 = − 20 − −6

−8 + −8 + 2 = − 20 + 6

− 14 = −14, true


5m − 2 m − 10 =

−25 5 − 5 − 2 −5 −

10 = −25 − 25 − −10 −

10 = −25 − 25 − −20 = −25 − 25 + 20 = −25

−5 = −25, false


8m − 3m − 5 = −40

8 7 − 3 7 − 5 = −40

− 21− 5 = −40

56 − 16 = −40

40 = −40, false


x 2

+ 6x + 8 =

0 − 4 2 + 6 −4 +

8 = 0 16 + − 24 + 8

= 0 24 + − 24 = 0

0 = 0, true


x 2

+ 6x + 8 =

0 − 2 2 + 6 −2 +

8 = 0 4 + − 12 + 8 =

0 12 + − 12 = 0

0 = 0, true



= 3y 2

+ y = −2

2

3 − 1 + −1 = −2

+

= −2 3 1 −1

3 +

= −2 −1

2 = −2,false The number is not a solution.

= 3y 2

+ 5y = −2

3 − 2 2

+ 5 − 2 =

−2 3 4 + 5 − 2 =

−2 12 + − 10 = −2

2 = −2, false


x + 3 x + 8 = x

− 4 + 3 − 4 + 8 = −4

−1 4 = −4

−4 = −4, true


x + 3 x + 8 = x

− 6 + 3 − 6 + 8 = −6

−3 2 = −6

−6 = −6, true


3z 3

− 12z = 0

3 − 23 − 12 − 2 =

0 3 − 8 − 12 − 2

= 0 − 24 − −24 = 0

− 24 + 24 = 0 0 = 0, true


3z 3

− 27z = 0

3 −33 − 27 −3 = 0

3 − 27 − 27 −3 = 0

−81− −81 = 0

−81+ 81 = 0 0 = 0, true




13 − 6 y = 3 y 2

+ 8 y − 11

13 − 6 −6 = 3 −6 2

+ 8 −6 − 11 − −36 = 3 36 + 8 − 6 − 11

13 + 36 = 108 + −48 + −11

49 = 108 + −59

49 = 49, true


4 + 3y = 2 y 2

+ 12 y − 1

+ 3 − 5 = 2 − 5 2 + 12 −5 − 1

4 + − 15 = 2 25 + 12 −5 − 1

− 11 = 50 +

− 60

+

−1

− 11 = −11, true


87. a. − 3x − 8

b. − 3x − 8 = −3 −5 − 8

= 15 − 8

= 7

88. a. −50 − 5

x

−50

− 5 = −50

− 5

−5 10 − 5

5

a. − 6 − 10x

− 6 − 10x = −6 − 10 −5

. −6 − −50

. −6 + 50

. 44

a. − 9 − 6x

− 9 − 6x = −9 − 6 −5

−9 − −30

−9 + 30

21

a. x + − 10 − x2

x + − 10 − x2

= −5 + − 10 − −52

− 15 − 25

−40

a. x + − 12 + x2

x + − 12 + x2

= − 5 + −12 + −52

− 17 + 25

8

a. e = − 2x 2

+ 57x + 143 = − 2 3 2 + 57 3 + 143

e = − 2 9 + 57 3 + 143

= − 18 + 171+ 143

e = 296 296 billion worldwide emails in 2010

e = − 2x 2

+ 57x + 143

e = − 2 2 2 + 57 2 +

143 e = − 2 4 + 57 2

+ 143 e = − 8 + 114 + 143

e = 249 249 billion worldwide emails in 2009

According to the model there were 296 billion –

249 billion, or 47 billion more worldwide emails

in 2010.

According to the graph there were 294 billion –

247 billion, or 47 billion more worldwide emails

in 2010. This is the same as the number obtained by the model.

a. D = − 40x 10 − x + 6310 D

= − 40 310 − 3 + 6310

D = − 40 3 7 + 6310

= − 840 + 6310

D = 5470 The mean credit card debt in 2011 was $5470.


Section 2.6 Solving Equations: The Addition and Multiplication Properties of Equality

D = − 40 x 10 − x + 6310 D

= − 40 5 10 − 5 + 6310

D = − 40 5 5 + 6310

D = − 1000 + 6310

D = 5310 The mean credit card debt in 2013 was $5310.

According to the model the mean credit card

balance in 2013 was $5470 – $5310, or $160 less than in 2011.

According to the model the mean credit card

balance in 2013 was $5476 – $5325, or $151 less than in 2011. The number obtained by

the model is $9 greater. – 97. Answers will vary.

does not make sense; Explanations will vary. Sample

explanation: –3 is not a solution because when it is substituted for the variable in the equation a false

statement occurs. makes sense

makes sense makes sense


A sample change is: − 14 ÷ 7 ⋅ 2 = − 2 ⋅ 2 = −4. false; Changes to make the statement true will vary. A

sample change is: − 2 6 − 4 2

3 = − 2 6 − 163

− 2 −103

− 2 −1000

2000 true true

2 2

− 12 ÷ − 4 = 4 − 12 ÷ −4

−8 ÷ −4

2 No additional parentheses are needed. 576

94 670

823

297

526

26 53 1383

106 323 318

5 1383 ÷ 53 = 26 R 5

− 6 = y − 15

− 6 = 9 − 15

− 6 = −6, true


8 + w = −12

+ − 20 = −12

−12 = −12, true


7 = w

−w

7 = 28

−28

7 = −1, false


2.6 Check Points

We can isolate the variable, x, by adding 5 to both sides of the equation.

− 5 = 12

− 5 + 5 = 12 + 5

+ 0 = 17

= 17 Check:

x − 5 = 12

17 − 5 = 12 12 = 12

The solution set is 17 .



We can isolate the variable, z, by subtracting 30 from both sides of the equation.

z + 30 = 20

z + 30 − 30 = 20 − 30

z = −10 Check:

z + 30 = 20

− 10 + 30 = 20

20 = 20

The solution set is −10 .

We can isolate the variable, y, by adding 13 to both

sides of the equation. − 9 = y − 13

− 9 + 13 = y − 13 + 13

4 = y

Check:

− 9 = 4 − 13

− 9 = 4 − 13

− 9 = −9


We can isolate the variable, w, by subtracting 12

from both sides of the equation.

12 + w = −14

12 + w − 12 = −14 − 12

w = −26 Check:

12 + w = −14

12 + −26 = −14

−14 = −14


We can isolate the variable, x, by multiplying both

sides of the equation by 5. = 12

3

3 ⋅ x

= 3 ⋅12 3

1x = 36

x = 36 Check: = 12

3 = 12

3 12 = 12


We can isolate the variable, w, by multiplying both sides of the equation by –6.

10 = w

−6

w − 6 ⋅ 10 = −6 ⋅

−6

− 60 = 1w

− 60 = w Check:

10 = w

−6

10 = −60

−6

10 = 10


a. We can isolate the variable, x, by dividing both

sides of the equation by 4. 4 x = 84

4 x = 84

4 1x = 21

= 21

Check:

4 x = 84

4 21 = 84

84 = 84


We can isolate the variable, y, by dividing both

sides of the equation by –11. − 11 y = 44

−11 y = 44

− 11 −11 1 y = −4

= −4

Check:

− 11 y = 44

− 11 −4 = 44 44 = 44




We can isolate the variable, z, by dividing both sides of the equation by 5. − 15 = 5z

−15 = 5z

5 −3 = 1z

−3 = z

Check: − 15 = 5z

− 15 = 5 −3

− 15 = −15


a. We can isolate the variable, x, by multiplying

both sides of the equation by –9. = −11

−9

− 9 ⋅ x = −9 ⋅

−11

−9

1x = 99

x = 99 Check: = −11

−9 = −11

−9 − 11 = −11


We can isolate the variable, y, by adding 12 to

both sides of the equation. y − 12 = −38

− 12 + 12 = −38 + 12

y = −26

Check: − 12 = −38

− 26 − 12 = −38

−38 = −38


We can isolate the variable, z, by dividing both sides of the equation by –8. 56 = −8z

= −8z

−8 −8 −

7 = 1z −

7 = z Check: = −8z

= −8 −7

56 = 56


We can isolate the variable, m, by subtracting 20

from both sides of the equation. 0 = m + 20

0 − 20 = m + 20 − 20

− 20 = m

Check:

0 = m + 20

= − 20 + 20

= 0


In the formula, A represents the child’s age, in

months. Thus, we substitute 50 for A. Then use the addition property of equality to find V, the number of words in the child’s vocabulary.

V + 900 = 60A

+ 900 = 60(50)

+ 900 = 3000

+ 900 − 900 = 3000 − 900

= 2100 At 50 months, a child will have a vocabulary of 2100 words.

Substitute 630 for R Then use the multiplication

property of equality to find L, the lifespan of mice. RL = 1890

630 L = 1890

630 L = 1890

630 1L = 3

L = 3

At 630 beats per minute, the average lifespan of mice is 3 years.




solving

equivalent

b c

subtract; solution

adding 7

subtracting 7

bc

divide

multiplying; 7

dividing; 8

2.6 Exercise Set

x − 4 = 19

− 4 + 4 = 19 + 4

= 23 Check:

− 4 = 19

23 − 4 = 19

19 = 19


x − 5 = 18

− 5 + 5 = 18 + 5

= 23 Check:

− 5 = 18

23 − 5 = 18

18 = 18


z + 8 = 12

+ 8 − 8 = 12 − 8

z = 4

Check: + 8 = 12

4 + 8 = 12

12 = 12


z + 13 = 15

+ 13 − 13 = 15 − 13

z = 2 Check: + 13 = 15

2 + 13 = 15

15 = 15

The solution set is 2 . z + 8 = −12

+ 8 − 8 = −12 − 8

z = −20 Check:

z + 8 = −12

− 20 + 8 = −12

− 12 = −12

The solution set is −20 . z + 13 = −15

+ 13 − 13 = −15 − 13

z = −28 Check:

z + 13 = −15

− 28 + 13 = −15

− 15 = −15


−2 = y + 14

− 2 − 14 = y + 14 − 14

−16 = y Check:

− 2 = y + 14

− 2 = −16 + 14

− 2 = −2

The solution set is −16 . −13 = y + 11

− 13 − 11 = y + 11 − 11

− 24 = y Check:

− 13 = y + 11

− 13 = − 24 + 11

− 13 = −13



− 17 = w − 5

− 17 + 5 = w − 5 + 5

−12 = w Check:

− 17 = w − 5

− 17 = − 12 − 5

− 17 = −17


− 21 = w − 4

− 21 + 4 = w − 4 + 4

−17 = w Check: − 21 = w − 4

− 21 = − 17 − 4

− 21 = −21

The solution set is −17 . − 6 + y = −20

− 6 + y + 6 = − 20 + 6

= −14 Check:

− 6 + y = −20

− 6 + 14 = −20 −20 = −20


− 8 + y = −29

− 8 + y + 8 = −29 +

8 y = −21

Check:

− 8 + y = −29

− 8 + − 21 = −29 −29 = −29


7 + x = 11

7 + x − 7 = 11 − 7

x = 4 Check:

7 + x = 11

7 + 4 = 11

11 = 11



18 + x = 14

18 + x − 18 = 14 − 18

x = −4 Check:

18 + x = 14

+ − 4 = 14

14 = 14


x

= 5 6

6 ⋅ x

= 6 ⋅ 5 6

1x = 30

x = 30 Check: = 5

6 = 5

6 5 = 5


x

= 4 7

7 ⋅ x

= 7 ⋅ 4 7

1x = 28

x = 28 Check: = 4

7 = 4

7 4 = 4




11 = y

−3

− 3 ⋅ 11 = −3

y

−3 −33 = 1y

−33 = y

Check:

11 = y

−3

11 = −33

−3

11 = 11


8 = y

−5

− 5 ⋅ 8 = −5

y

−5 − 40 = 1y

− 40 = y

Check:

8 = y

−5

= −40 8

−5 8 = 8


5 z = 35

5 z = 35 5 5 1z = 7

z = 7

Check:

5 z = 35

5 7 = 35

35 = 35


6 z = 42

6 z = 42 6 6 1z = 7 z = 7

Check:

6 z = 42

6 7 = 42

42 = 42

The solution set is

− 7 y = 63

−7 y = 63

− 7−7 1

y = −9

y = −9 Check:

− 7 y = 63

− 7 − 9 = 63 63 = 63

The solution set is

− 4 y = 32

−4 y = 32 − 4 −4

1 y = −8

= −8 Check:

−4 y = 32

− 4 −8 = 32 32 = 32

The solution set is

− 48 = 8x

−48 = 8x

8 − 6 = 1x

− 6 = x

Check: − 48 = 8x

− 48 = 8 −6

− 48 = −48

The solution set is

7 .

−9 .

−8 .

−6 .


− 56 = 8x

−56 = 8x

8 − 7 = 1x

− 7 = x

Check: − 56 = 8x

− 56 = 8 −7

− 56 = −56

The solution set is

− 18 = −3z

−18 = −3z − 3 −3

= 1z

= z Check:

− 18 = −3z

− 18 = −3 6

− 18 = −18

The solution set is

− 54 = −9z

−54 = −9z −9 −9

= 1z

= z Check:

− 54 = −9z

− 54 = −9 6

− 54 = −54

The solution set is

− 17 y = 0

−17 y = 0

− 17 −17 1 y = 0

y = 0

Check: − 17 y = 0

− 17 0 = 0

0 = 0

The solution set is

−7 .

6 .

6 .

0 .


− 16 y = 0

−16 y = 0

− 16 −16 1 y = 0

y = 0

Check: − 16 y = 0

− 16 0 = 0

0 = 0


x − 7 = −14

− 7 + 7 = −14 +

7 x + 0 = −7

x = −7 Check:

x − 7 = −14

− 7 − 7 = −14

−14 = −14


x − 8 = −16

− 8 + 8 = − 16 + 8

+ 0 = −8

= −8 Check:

− 8 = −16

− 8 − 8 = −16

− 16 = −16


− 7 x = −14 −7 x

= −14 −7

−7 1x = 2 x = 2

Check:

−7 x = −14

− 7 2 = −14

−14 = −14




− 8 x = −16

−8 x =

−16 −8 −8 1x = 2

= 2 Check:

−8 x = −16

− 8 2 = −16

− 16 = −16


x = −14 −7

− 7 ⋅ x = − 7 ⋅ −14

−7 1x = 98

x = 98

Check: = −14

−7 −98 = −

14 −7 −14 = −14


x = −16 −8

− 8 ⋅ x = − 8 ⋅ −16

−8 1x = 128

= 128

Check: = −16

−8 = −16

−8 −16 = −16


− 14 = 7 + x

− 14 − 7 = 7 + x − 7

−21 = x

Check:

− 14 = 7 + x

− 14 = 7 + −21

− 14 = −14


−16 = 8 + x

− 16 − 8 = 8 + x − 8

−24 = x Check: − 16 = 8 + x

− 16 = 8 + −24

− 16 = −16


y − 172 = −243

− 172 + 172 = −243 + 172

y = −71

Check:

− 172 = −243

− 71 − 172 = −243

−243 = −243


y − 183 = −421

− 183 + 183 = − 421 + 183

y = −238 Check:

y − 183 = −421

− 238 − 183 = −421

− 421 = −421

The solution set is −238 . 96 = x − 128

96 + 128 = x − 128 + 128

224 = x Check:

= x − 128

= 224 − 128

= 96

The solution set is 224 . 84 = x − 137

+ 137 = x − 137 + 137

221 = x Check:

= x − 137

= 221 − 137

= 84

The solution set is 221 . 96 Copyright © 2017 Pearson Education, Inc.

− 5 w = 1015

−5 w = 1015

− 5 −5 w =

−203 Check: − 5 w = 1015 − 5

−203 = 1015

1015 = 1015 The

solution set is

− 6 w = 1812

−6 w = 1812 − 6 −6

=

−302 Check: −6 w = 1812

− 6 −302 =

1812 1812 = 1812

The solution set is

− 496 = −31z

−496 = −31z

−31 −31 16 = z

Check:

− 496 = −31z

− 496 = −31 16

− 496 = −496 The

solution set is

− 714 = −42z

−714 = −42z

−42 −42 17 = z

Check:

− 714 = −42z

− 714 = −42 17

− 714 = −714 The

solution set is

−203 .

−302 .

16 .

17 .


−496 = − 31+ z

− 496 + 31 = −31 + z + 31

− 465 = z Check:

− 496 = − 31+ z

− 496 = − 31 + −465

− 496 = −496


− 714 = − 42 + z

− 714 + 42 = − 42 + z + 42

− 672 = z

Check:

− 714 = − 42 + z

− 714 = − 42 + −672

− 714 = −714


0 = 31+ z

0 − 31 = 31 + z − 31

−31 = z Check:

= 31+ z

= 31 + −31

= 0


0 = 42 + z

− 42 = 42 + z − 42

− 42 = z Check: 0 = 42 + z

= 42 + −42

= 0




z = −496 31

31 ⋅ z

= 31 −496 31

= −15, 376 Check:

= −496

31

−15, 376 = −496 31 − 496 = −496

The solution set is −15, 376 .

z = −714 42

42 ⋅ z

= 42 −714 42

= −29, 988

Check: = −714

42

z = −714 −42

− 42 ⋅ z =

−42

−714

−42 z = 29,988

Check:

= −714

−42 29,988 = −

714 −42 − 714 = −714

The solution set is 29,988 . x − =

− + = +

= + x + =

+ − = −

= −

51.

98

−29, 988 = −714 42 − 714 = −714

The solution set is −29, 988 .

= −496

−31

− 31 ⋅ z =

− 31

−496

−31 = 15, 376

Check: = −496

−31 15, 376 = −

496 −31 −496 = −496

The solution set is 15, 376 .

55. x

=

⋅ x

= ⋅

56. = x

= x

= x

57. x − 12 = −8

x = − 8 + 12

x = 4 The number is 4.

58. x − 23 = −8

x − 23 + 23 = − 8 + 23

x = 15

The number is 15.

59. x + 50 = 35

x + 50 − 50 = 35 − 50

x = −15 The number is –15.

Copyright © 2017 Pearson Education, Inc.

Section 2.6

60. x + 60 = 25

x + 60 − 60 = 25 − 60

x = −35

The number is –35.

61. x = −20

−2 − 2 ⋅ x = −2 −20

−2 x = 40

The number is 40.

x = −15 −3

− 3 ⋅ x = − 3 −15

−3 x = 45

The number is 45. 9 x = −63

9 x = −63

9 = −7

The number is –7. 8 x = −48

8 x = −48

8 = −6

The number is –6. S = 1850, M = 150

C + M = S

+ 150 = 1850

= 1850 − 150

= 1700 The cost of the computer is $1700.

C = 520, S = 650

C + M = S

520 + M = 650

= 650 − 520

= 130

The markup is $130.

Solving Equations: The Addition and Multiplication Properties of Equality

10D − 335 = 7x

10 65 − 335 = 7x

650 − 335 = 7x

= 7x

= 7x

7 45 = x

45 years after 1980, or in 2025 the U.S. diversity index will be 65.

10D − 335 = 7x

10 72 − 335 = 7x

720 − 335 = 7x

= 7x

= 7x

7 55 = x

55 years after 1980, or in 2035 the U.S. diversity index will be 72.

M = n

5

2 = n

5

5 2 n = 5

5

10 = n If you are 2 miles away from the lightning flash, it will take 10 seconds for the sound of thunder to reach you.

M = n

5

3 = n

5

n

5 3 = 5

5

15 = n If you are 3 miles away from the lightning flash, it will take 15 seconds for the sound of thunder to

reach you.

71. B

72. C



a. A = lw 171 =

l 9

= l 9

9 = l

The length is 19 yards.

P = 2l + 2w

P = 2 19 + 2

9 P = 38 + 18

P = 56 The perimeter is 56 yards.

a. A = lw

161 = l 7

= l 7

7 23 = l

The length is 23 feet.

P = 2l + 2w

P = 2 23 + 2

7 P = 46 + 14

P = 60 The perimeter is 60 feet.

150 p = wA

150 p = 60 25

150 p = 1500

150 p = 1500

150 p = 10

The dosage is 10 milligrams.

150 p = wA

150 p = 90 25

150 p = 2250

150 p = 2250

150 p = 15

The dosage is 15 milligrams.


does not make sense; Explanations will vary. Sample

explanation: They are both mathematically the same.

82. makes sense 83. makes sense 84. does not make sense; Explanations will vary.

Sample explanation: To isolate the variable we

need to use the multiplication property of equality. 85. false; Changes to make the statement true will vary.

A sample change is: If − 3 x = 18, then x = 18

. −3

true true


A sample change is: y = −7.

Answers will vary. Example: x − 100 = −101 Answers will vary. Example: − 60 x = −120 x − 37, 256 = −19,125

− 37, 256 + 37, 256 = − 19,125 + 37, 256

= 18,131

The solution set is 18,131 .

67, 592 = y + 127,963

67, 592 − 127, 963 = y + 127,963 −

127,963 − 60,371 = y

The solution set is −60,371 .

w

= −3002 578

578 ⋅ w

= −3002 578 578

= −1, 735,156

The solution set is −1, 735,156 .

− 860, 778 = −1746z

−860, 778 =

−1746z −1746−1746 493 = z


− 10 2 = − 10 − 10 = 100

− 10 2 = − 10 ⋅ 10 = −100


Chapter 2 Review Exercises

x 3

− 4 x = −13 − 4 −1

−1 − −4 −1 + 4

3

a = 7; b = 19

a = − 4; b = 13

a = − 3; b = −10


1.

2.

− 93 < 17

− 2 > −200

0 > −1

−58 = 58

− 58 = −58



− − 72 = 72

− − 30 = −30

− − 63 = 63 = 63

− 6 + 8 = 2

− 23 + ( − 17) = −40

18 + ( − 25) = −7

− 15 + 29 = 14

326 + ( −326) = 0



7 + − 5 + −13 + 4 = 7 + 4 + − 5 + −13

11 + −18

−7

− 41 + 213 + −15 + − 72 = 213 + −41 + −15 + −72

= 213 + −128

= 85

− 1312 + 512 = −800 The person is standing at 800 feet below sea level.

25 + − 3 + 2 + 1 + − 4 + 2 = 25 + 2 + 1 + 2 + −3 + −4

30 + −7

23 The water level is 23 feet.

9 − 13 = 9 + ( −13)

− 9 − ( − 15) = − 9 + 15 = 6

2 − 20 = 2 + ( − 20) = −18

− 28 − 31 = −28 + ( − 31) = −59

146 − − 204 = 146 + 204 = 350

− 124 − ( −59) = −124 + 59 = −65

− 75 − ( −75) = − 75 + 75 = 0

75 − ( −75) = 75 + 75 = 150

0 − ( − 83) = 0 + 83 = 83

− 7 − ( −5) + 11 − 16 = −7 + 5 + 11 − 16 5 + 11 + ( −7) + −16

16 + ( −23)

−7

− 25 − 4 − − 10 + 16 = −25 + − 4 + 10 + 16

−29 + 16

−3

39 − ( −11) = 39 + 11 = 50

− 50 − 30 = − 50 + −30 = −80

− 50 − 30 = − 50 + −30 = −80


− 20 − 80 = − 20 + −80 = −100

26, 500 − −650 = 26,500 + 650 = 27,150

The difference in elevation is 27,150 feet. − 7( −12) = 84

14( −5) = −70

− 11 6 = −66

( − 7)( −2)(10) = 140

5( − 2)( −3)( − 4) = −120

− 6 2 = −6 −6 = 36

− 6 2 = − 6 ⋅ 6 = −36

− 5 3 = −5 −5 −5 = −125

− 2 4 = − 2 −2 −2 −2 = 16

− 2 4 = − 2 ⋅ 2 ⋅ 2 ⋅ 2 = −16

− 3 5 = −3 − 3 − 3 −3 − 3 = −243

45

= −9

−5

− 90 ÷ 9 = −10

− 44 ÷ − 4 = 11

0 ÷ − 50 = 0

53. − 50 ÷ 0 undefined

54. −1506 = −502

3

55. −221

= 17 −13

a. Lost the most money in May.

− 190 12 = −2280 The total loss would be $2280.


Lost money in January, April, and May.

− 150 + −75 + − 190 = −415 The total loss was $415.

Made money in February, March, and

June. 15 + 130 + 30 = 175

The total made was $175.

−415 + 175 = −40 6

The mean loss was $40.

− 150 − −75 = −75

You loss $75 more in January.

3 −150 = −450

The total loss would be $450.

− 40 ÷ 5 ⋅ 2 = − 8 ⋅ 2 −16

− 6 + −2 ⋅ 5 = −6 + −10

−16

30 ÷ 10 − 2 3 = 30 ÷ 10 −8

3 −8

−24

− 12 − 3 − 4 ⋅ 5 = −12 − 3 − 20

−12 − −17

−12 + 17

5

16 − 30 ÷ 10 − 4 2 − 6 = 16 − 30 ÷ 10(16) − 6

16 − 3 16 − 6

16 − 48 − 6

−38

28 ÷ (2 − 4 2

) = 28 ÷ (2 − 16)

28 ÷ ( −14) −2

24 − 36 ÷ 4 ⋅ 3 − 1 = 24 − 9 ⋅ 3 − 1 24 − 27 − 1

24 + −27 + −1

24 + −28

−4



− 8 −4 2 − 5 + 5 −3 = −8 − 4 −3 + 5 −3

= −8

12 + −15

= −8 −3

= 24

65.

6 − 10 + 3

=

6 −7

2 −15 − 9 − 3 −30 − −27 = −42

−30 + 27 −42

−3 14

− 10 − 12 − 14 = −22 − 14

22 − 14

8

− x 2 − 3 x + 4 x + 2 = − −5 2 − 3 − 5 + 4 −5 + 2

−25 − 3 −5 + 4 −3

−25 − −15 + −12

−25 + 15 + −12

15 + −37

−22

b 2

− 4 ac = −5 2

− 4 3 −2

25 − 4 3 −2

25 − −24

25 + 24

49

5 x + 16 = − 8 − x

5 − 6 + 16 = − 8 − −6

− 30 + 16 = − 8 + 6

− 14 = −2,false The number is not a solution.

2 x + 3 − 18 = 5x

2 − 4 + 3 − 18 = 5 −4

2 − 1 − 18 = −20 −2 − 18 = −20

− 20 = −20, true


a. F = − x 2

− x + 2860 + x 22x + 152

= − 10 2 − 10 + 2860 + 10 22 ⋅ 10 +

152 F = − 10 2

− 10 + 2860 + 10 220 +

152 F = − 10 2

− 10 + 2860 + 10 372

= − 100 − 10 + 2860 + 3720

= − 110 + 6580

= 6470 There was 6470 farmers markets in 2010.

The model overestimates by 6470 – 6132, or by 338 farmers markets.

x − 10 = 22

− 10 + 10 = 22 + 10

= 32 Check:

− 10 = 22

32 − 10 = 22

22 = 22

The solution set is 32 . −14 = y + 8

− 14 − 8 = y + 8 − 8

−22 = y Check:

− 14 = y + 8

− 14 = − 22 + 8

− 14 = −14


z = 10 6

6 ⋅ z

= 6 ⋅10 6

1x = 60

x = 60 Check: = 10

6 = 10

6 10 = 10



7 = w

−8

− 8 ⋅ 7 = − 8⋅ w

−8

−56 = w Check:

7 = w

−8

7 = −56 −8

7 = 7


7 x = 77

7 x = 77

7 = 11

Check:

7 x = 77

7 11 = 77

77 = 77


− 36 = −9y

−36 = −9 y − 9 −9

4 = y Check:

− 36 = −9 y

− 36 = −9 4

− 36 = −36


a. rt = 420

30t = 420

30t = 420

30

t = 14

The time for the trip is 14 hours.

rt = 420

60t = 420

60t = 420

60 t = 7

The time for the trip is 7 hours.

The difference is 14 hours – 7 hours, or 7 hours.

Chapter 2 Test

Chapter 2 Test

14 − ( − 26) = 14 + 26 = 40

− 9 + 3 + − 11 + 6 = − 9 + −11 + 3 + 6

− 20 + 9

−11

− 3( − 17) = 51

2( −4)( − 5)( − 1) = −40

− 50 ÷ 10 = −5

− 6 − 5 − 12 = −6 − −7

−6 + 7

1

− 3 −4 ÷ 7 − 10 = −3 − 4 ÷ −3

= 12 ÷ −3

= −4

8. 6 − 8 2

5 − 7 3 = − 2

2 −23

= 4 −8 = −32

9.

3 − 2 − 2 2

=

−6 − 4

− 2 8 − 3 −2 5 −10

−10 1

− 1 > −100

16, 200 − − 830 = 16,200 + 830 = 17,030 The

difference in elevation is 17,030 feet.

a. − 7 = 7

− − 7 = −7

− −7 = 7



− x 2 − 5x + 2 x + 3 = − −4 2 − 5 −4 + 2 −4 + 3

−16 − 5 − 4 + 2 −1

−16 − − 20 + −2

−16 + 20 + −2

20 + −18

2

3 x + 2 − 15 = 4x

3 − 9 + 2 − 15 = 4 −9

3 −7 − 15 = −36 −21− 15 = −36

− 36 = −36, true


x − 12 = 25

− 12 + 12 = 25 + 12

= 37 Check:

− 12 = 25

37 − 12 = 25

25 = 25


− 70 = −7 y

−70 = −7

y −7 −7 10 = y

Check:

− 70 = −7 y

− 70 = −7 10

− 70 = −70


− 16 = y + 7

− 16 − 7 = y + 7 − 7

−23 = y Check:

− 16 = y + 7

− 16 = − 23 + 7

− 16 = −16


w

= 6

−5

− 5⋅ w

= − 5⋅ 6 −5 = −30

Check: = 6

−5

−30 = 6 −5

6 = 6

The solution set is −30 . a. A = − 5x + 308

= − 5 20 + 308

= − 100 + 308

A = 208 The office area per worker in 2010 was 208 square feet.

The model underestimates by 225 square feet

– 208 square feet, or by 17 square feet.


€¦ · solution manual for developmental mathematics 1st edition by blitzer link full download

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